Chain Theorems of Lines Circles and Planes

In this paper we will prove Clifford chain theorem for general lines in the plane by using real cross ratio lemma. We will then discuss the Clifford chain theorem for degenerate lines and will obtain 6 types of Clifford figures with examples. We will also find that the Clifford chain theorem fails for some cases and we will show some examples. At the end we will obtain three similar chain theorems for circles and planes by applying Clifford chain theorem for general lines in the plane under the transformations of stereographic projection and circle inversion. These three chain theorems are (1): chain theorem for general circles on the sphere, (2): chain theorem for general circles in the plane and (3): chain theorem for general planes in the space. We will also prove Miquel's pentagon theorem by applying Clifford chain theorem for four general lines

Observation of Multi-Tev Gamma Rays from the Crab Nebula Using the Tibet Air Shower Array

The Tibet experiment, operating at Yangbajing (4,300 m above sea level), is the lowest energy air shower array and the new high density array constructed in 1996 has sensitivity to $\gamma$-ray air showers at energies as low as 3 TeV. With this new array, the Crab Nebula was observed in multi-TeV $\gamma$-rays and a signal was detected at the 5.5 $\sigma$ level. We also obtained the energy spectrum of $\gamma$-rays in the energy region above 3 TeV which partially overlaps those observed with imaging atmospheric Cherenkov telescopes. This is the first observation of $\gamma$-ray signals from point sources with a conventional air shower array using scintillation detectors.Comment: 9 pages, 4 figures, Accepted for publication in ApJ Letter

Detection of Multi-TeV Gamma Rays from Markarian 501 during an Unforeseen Flaring State in 1997 with the Tibet Air Shower Array

In 1997, the BL Lac Object Mrk 501 entered a very active phase and was the brightest source in the sky at TeV energies, showing strong and frequent flaring. Using the data obtained with a high density air shower array that has been operating successfully at Yangbajing in Tibet since 1996, we searched for gamma-ray signals from this source during the period from February through August in 1997. Our observation detected multi-TeV $\gamma$-ray signals at the 3.7-Sigma level during this period. The most rapid increase of the excess counts was observed between April 7 and June 16 and the statistical significance of the excess counts in this period was 4.7-Sigma. Among several observations of flaring TeV gamma-rays from Mrk 501 in 1997, this is the only observation using a conventional air shower array. We present the energy spectrum of gamma-rays which will be worthy to compare with those obtained by imaging atmospheric Cerenkov telescopes.Comment: 9 pages, 7 figures, To appear in Ap

The energy spectrum of all-particle cosmic rays around the knee region observed with the Tibet-III air-shower array

We have already reported the first result on the all-particle spectrum around the knee region based on data from 2000 November to 2001 October observed by the Tibet-III air-shower array. In this paper, we present an updated result using data set collected in the period from 2000 November through 2004 October in a wide range over 3 decades between $10^{14}$ eV and $10^{17}$ eV, in which the position of the knee is clearly seen at around 4 PeV. The spectral index is -2.68 $\pm$ 0.02(stat.) below 1PeV, while it is -3.12 $\pm$ 0.01(stat.) above 4 PeV in the case of QGSJET+HD model, and various systematic errors are under study now.Comment: 12 pages, 7 figures, accepted by Advances in space researc

Primary proton spectrum between 200 TeV and 1000 TeV observed with the Tibet burst detector and air shower array

Since 1996, a hybrid experiment consisting of the emulsion chamber and burst detector array and the Tibet-II air-shower array has been operated at Yangbajing (4300 m above sea level, 606 g/cm^2) in Tibet. This experiment can detect air-shower cores, called as burst events, accompanied by air showers in excess of about 100 TeV. We observed about 4300 burst events accompanied by air showers during 690 days of operation and selected 820 proton-induced events with its primary energy above 200 TeV using a neural network method. Using this data set, we obtained the energy spectrum of primary protons in the energy range from 200 to 1000 TeV. The differential energy spectrum obtained in this energy region can be fitted by a power law with the index of -2.97 $\pm$ 0.06, which is steeper than that obtained by direct measurements at lower energies. We also obtained the energy spectrum of helium nuclei at particle energies around 1000 TeV.Comment: 25 pages, 22 figures, Accepted for publication in Phys. Rev.

4.5 years multi-wavelength observations of Mrk 421 during the ARGO-YBJ and Fermi common operation time

We report on the extensive multi-wavelength observations of the blazar Markarian 421 (Mrk 421) covering radio to gamma-rays, during the 4.5 year period of ARGO-YBJ and Fermi common operation time, from August 2008 to February 2013. In particular, thanks to the ARGO-YBJ and Fermi data, the whole energy range from 100 MeV to 10 TeV is covered without any gap. In the observation period, Mrk 421 showed both low and high activity states at all wavebands. The correlations among flux variations in different wavebands were analyzed. Seven large flares, including five X-ray flares and two GeV gamma-ray flares with variable durations (3-58 days), and one X-ray outburst phase were identified and used to investigate the variation of the spectral energy distribution with respect to a relative quiescent phase. During the outburst phase and the seven flaring episodes, the peak energy in X-rays is observed to increase from sub-keV to few keV. The TeV gamma-ray flux increases up to 0.9-7.2 times the flux of the Crab Nebula. The behavior of GeV gamma-rays is found to vary depending on the flare, a feature that leads us to classify flares into three groups according to the GeV flux variation. Finally, the one-zone synchrotron self-Compton model was adopted to describe the emission spectra. Two out of three groups can be satisfactorily described using injected electrons with a power-law spectral index around 2.2, as expected from relativistic diffuse shock acceleration, whereas the remaining group requires a harder injected spectrum. The underlying physical mechanisms responsible for different groups may be related to the acceleration process or to the environment properties.Comment: 17 pages, 9 figures, 5 tables, Accepted for publication in ApJ

The cosmic ray proton plus helium energy spectrum measured by the ARGO-YBJ experiment in the energy range 3-300 TeV

The ARGO-YBJ experiment is a full-coverage air shower detector located at the Yangbajing Cosmic Ray Observatory (Tibet, People's Republic of China, 4300 m a.s.l.). The high altitude, combined with the full-coverage technique, allows the detection of extensive air showers in a wide energy range and offer the possibility of measuring the cosmic ray proton plus helium spectrum down to the TeV region, where direct balloon/space-borne measurements are available. The detector has been in stable data taking in its full configuration from November 2007 to February 2013. In this paper the measurement of the cosmic ray proton plus helium energy spectrum is presented in the region 3-300 TeV by analyzing the full collected data sample. The resulting spectral index is $\gamma = -2.64 \pm 0.01$. These results demonstrate the possibility of performing an accurate measurement of the spectrum of light elements with a ground based air shower detector.Comment: 18 pages, 8 figures, preprint submitted to Phys. Rev.

EAS age determination from the study of the lateral distribution of charged particles near the shower axis with the ARGO-YBJ experiment

The ARGO-YBJ experiment, a full coverage extensive air shower (EAS) detector located at high altitude (4300 m a.s.l.) in Tibet, China, has smoothly taken data, with very high stability, since November 2007 to the beginning of 2013. The array consisted of a carpet of about 7000 m$^2$ Resistive Plate Chambers (RPCs) operated in streamer mode and equipped with both digital and analog readout, providing the measurement of particle densities up to few particles per cm$^2$. The unique detector features (full coverage, readout granularity, wide dynamic range, etc) and location (very high altitude) allowed a detailed study of the lateral density profile of charged particles at ground very close to the shower axis and its description by a proper lateral distribution function (LDF). In particular, the information collected in the first 10 m from the shower axis have been shown to provide a very effective tool for the determination of the shower development stage ("age") in the energy range 50 TeV - 10 PeV. The sensitivity of the age parameter to the mass composition of primary Cosmic Rays is also discussed

Study of the diffuse gamma-ray emission from the Galactic plane with ARGO-YBJ

The events recorded by ARGO-YBJ in more than five years of data collection have been analyzed to determine the diffuse gamma-ray emission in the Galactic plane at Galactic longitudes 25{\deg} < l < 100{\deg} and Galactic latitudes . The energy range covered by this analysis, from ~350 GeV to ~2 TeV, allows the connection of the region explored by Fermi with the multi-TeV measurements carried out by Milagro. Our analysis has been focused on two selected regions of the Galactic plane, i.e., 40{\deg} < l < 100{\deg} and 65{\deg} < l < 85{\deg} (the Cygnus region), where Milagro observed an excess with respect to the predictions of current models. Great care has been taken in order to mask the most intense gamma-ray sources, including the TeV counterpart of the Cygnus cocoon recently identified by ARGO-YBJ, and to remove residual contributions. The ARGO-YBJ results do not show any excess at sub-TeV energies corresponding to the excess found by Milagro, and are consistent with the predictions of the Fermi model for the diffuse Galactic emission. From the measured energy distribution we derive spectral indices and the differential flux at 1 TeV of the diffuse gamma-ray emission in the sky regions investigated.Comment: 11 pages, 6 figures, published in AP